Theorems · Definition · global analysis

HasStrictFDerivAt.implicitFunctionDataOfProdDomain

{𝕜 : Type u_1} →
  [inst : NontriviallyNormedField 𝕜] →
    {E₁ : Type u_2} →
      [inst_1 : NormedAddCommGroup E₁] →
        [inst_2 : NormedSpace 𝕜 E₁] →
          [inst_3 : CompleteSpace E₁] →
            {E₂ : Type u_3} →
              [inst_4 : NormedAddCommGroup E₂] →
                [inst_5 : NormedSpace 𝕜 E₂] →
                  [inst_6 : CompleteSpace E₂] →
                    {F : Type u_4} →
                      [inst_7 : NormedAddCommGroup F] →
                        [inst_8 : NormedSpace 𝕜 F] →
                          [inst_9 : CompleteSpace F] →
                            {u : E₁ × E₂} →
                              {f : E₁ × E₂ → F} →
                                {f'u : E₁ × E₂ →L[𝕜] F} →
                                  HasStrictFDerivAt f f'u u →
                                    (f'u ∘SL ContinuousLinearMap.inr 𝕜 E₁ E₂).IsInvertible →
                                      ImplicitFunctionData 𝕜 (E₁ × E₂) F E₁

Given f : E₁ × E₂ → F strictly differentiable at u with invertible partial derivative f₂u : E₂ →L[𝕜] F, we may construct an ImplicitFunctionData 𝕜 (E₁ × E₂) F E₁ with f as its leftFun and Prod.fst : E₁ × E₂ → E₁ as its rightFun by proving that the kernels of the associated leftDeriv and rightDeriv are complementary.

Defined in

Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain

Cited by

8 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpaceNormedAddCommGroupNormedSpaceCompleteSpace